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2.5B Relationship of n, all zeros, real zeros & x-intercepts

2.5B Relationship of n, all zeros, real zeros & x-intercepts. N = degree of polynomial (biggest power) ALL zeros (solutions) = there are n of them including real, imaginary and repeated

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2.5B Relationship of n, all zeros, real zeros & x-intercepts

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  1. 2.5B Relationship of n, all zeros, real zeros & x-intercepts • N = degree of polynomial (biggest power) • ALLzeros (solutions) = there are n of them including real, imaginary and repeated • REAL zeros = only the solutions that are NOT imaginary. We can SEE them (where they touch the x-axis) • X-intercepts = Where the graph touches the x-axis (these are the REAL zeros)

  2. Zeros that occur in PAIRS • Some zeros ALWAYS occur in conjugate PAIRS • Complex Zeros • 2i and -2i Factors (x-2i) (x+2i) • -2 + 5i and -2 – 5i Factors (x+2+5i)(x+2-5i) • Irrational (radical) Zeros • and - Factors (x-)(x+) • -3 + and -3 - Factors (x+3+)(x+3-)

  3. Find a polynomial given some zeros • Write EACH zero as a factor • If the zero is imaginary or a radical, write a factor for it AND its CONJUGATE. (Always occur in pairs) • Multiply factors out (Foil conjugate pairs, extended distribute) • If a solution point is given, put result in ( ) with an “a” out front and find “a” by substituting in the x and y values of the point.

  4. Examples: • 1. Find a polynomial with the zeros 2,2,4-i and a solution point f(1) = -20

  5. Product of Linear Factors Irreducible… • Product of linear factors = Factors for the solutions written in a horizontal (linear) form. • ( )( )( )( ) • Irreducible over the rationals: (only integers) • NO imaginary values • NO radical values • Irreducible over the reals: (only reals) • NO imaginary values (Radical values are OK!!!)] • Completely factored: (Everything) n of them • One for EACH solution (Radicals and imaginaries OK!!)

  6. Examples: • 2. f(x) = - - 3x² + 12x – 18 Write the polynomial as the product of linear factors that are a) irreducible over the RATIONALS b) irreducible over the REALS and c) completely factored. (Hint: One factor is x² - 6)

  7. Use one given zero to find ALL zeros • If the zero given is imaginary or radical write 2 factors (one for each conjugate pair) • Multiply these conjugate pairs • Divide the original by this factor (Synthetic division if n=1, otherwise use long division) • Solve the new quotient by most appropriate method to find remaining zeros. (Total = n) • Write as a product of linear factors if asked.

  8. Examples: • 3. Find ALL the zeros of f(x)=- 7x²- x + 87 if one zero is 5 + 2i.

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